Method for Generating Constant Modulus Multi-Dimensional Modulations for Coherent Optical Communications

1. A computer-implemented method for generating a code for modulating signals transmitted for coherent optical communications, using a processor in connection with a memory, comprising: providing codewords represented by points in a constellation of the code into the memory; projecting the points in the constellation of the code onto a hyper-sphere by normalizing a sum of power over multiple dimensions to produce a multi-dimensional constellation, wherein the code is L-dimensional N-ary amplitude M-ary phase-shift keying including 4D-2A8PSK and 4D-2AQPSK, wherein amplitudes and phase offsets of the code are optimized, and wherein a sum of powers over L dimensions is constant for all codewords in the code; identifying point pairs having a smallest distance in the multi-dimensional constellation that limits a performance of the code; adjusting the point pairs in the multi-dimensional constellation to improve the performance of the code; and iterating the identifying and the adjusting until convergence, wherein the steps are performed in the processor.

2. The method of claim 1 , further comprising: optimizing positions of the point pairs in the multi-dimensional constellation using a performance criterion, wherein the performance criterion maximizes a minimum Euclidean distance, minimizes a bit-error rate, minimizes a union bound, maximizes mutual information, maximizes a reach along a nonliner fiber, minimizes a required signal-to-noise ratio, maximizes a tolerance to phase noise, or minimizes fluctuation of a power envelope in a signal transition and combinations thereof.

3. The method of claim 1 , wherein an initial code is selected from a group consisting of Gray-coded, dual-polarization M-ary phase-shift keying (DP-MPSK), M-ary quadrature-amplitude modulation (DP-MQAM), M-ary amplitude phase-shift keying (DP-MAPSK), and circular QAM formats, where M is an arbitrary positive integer depending on a specific target of data rates.

4. The method of claim 1 , wherein the code uses a subspace of a unitary space-time block code of any number of dimensions to be constant modulus, using a discrete Fourier transform, an exponential mapping, a Cayley transform, a Reed-Muller operator, Grassmann packing, or cyclic rotations.

5. The method of claim 1 , wherein the constellation is irregular adjusted by a meta-heuristic optimization procedure, nonlinear programming, or a gradient search method.

6. The method of claim 1 , further comprising: optimizing the multi-dimensional constellation while taking a transmit filter impulse response into account to generate near-constant modulus waveform over multiple symbols even for symbol transitions, by determining a power envelope for each codeword of adjacent blocks in the code.

7. The method of claim 6 , wherein codebook for the code changes over time using unitary rotation to minimize fluctuations of the power envelope.

8. The method of claim 6 , wherein an impulse response of a transmit filter is a convolution of all components affecting input to a fiber, including an electrical filter, such as root-raised-cosine filter, a response of an analog-to-digital converter, a response of an electro-optic driver or an optical filter, such as super-Gaussian filter.

9. The method of claim 6 , wherein a symbol timing of 4D carrier fields are synchronized.

10. The method of claim 6 , wherein a symbol timing of 4D carrier fields is unsynchronized and timing offsets between in-phase, quadrature, x-polarization and y-polarization are optimized to achieve near-constant modulus waveform.

11. The method of claim 1 , wherein the multiple dimensions include an in-phase component, a quadrature component, x-polarization, y-polarization, multiple symbols in a time domain, multiple channels in super-channel transmissions, multiple wavelengths in wavelength-division multiplexing, multiple modes in mode-division multiplexing, multiple cores in space-division multiplexing, multiple fibers, and combinations thereof.

12. The method of claim 1 , wherein the steps are performed offline.

13. A computer-implemented method for generating a code for modulating signals transmitted for coherent optical communications, using a processor in connection with a memory, comprising: providing codewords represented by points in a constellation of the code into the memory; projecting the points in the constellation of the code onto a hyper-sphere by normalizing a sum of power over multiple dimensions to produce a multi-dimensional constellation, wherein the code uses a subspace of a unitary space-time block code of any number of dimensions to be constant modulus, using a discrete Fourier transform, an exponential mapping, a Cayley transform, a Reed-Muller operator, Grassmann packing, or cyclic rotations; identifying point pairs having a smallest distance in the multi-dimensional constellation that limits a performance of the code; adjusting the point pairs in the multi-dimensional constellation points to improve the performance of the code; and iterating the identifying and the adjusting until convergence, wherein the steps are performed in the processor.

14. A computer-implemented method of generating a code for modulating signals transmitted for coherent optical communications, using a processor in connection with a memory, comprising: providing codewords represented by points in a constellation of the code into the memory; projecting the points in the constellation of the code onto a hyper-sphere by normalizing a sum of power over multiple dimensions to produce a multi-dimensional constellation; identifying point pairs having a smallest distance in the multi-dimensional constellation that limits a performance of the code; adjusting the point pairs in the multi-dimensional constellation points to improve the performance of the code; iterating the identifying and the adjusting until convergence; and optimizing the multi-dimensional constellation while taking a transmit filter impulse response into account to generate near-constant modulus waveform over multiple symbols even for symbol transitions, by determining a power envelope for each codeword of adjacent blocks in the code, wherein the steps are performed in the processor.